the-rule-of-31

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Note the relationship between the “A” and the “C#”. Here we can mathematically observe the imperceptible discrepancy with a pure major third. The pure third would have a frequency of 275 Hz, but the 31-division only misses the mark by .1 Hz, that is, it would only beat once every ten seconds (far longer than any lute string would resonate). Similarly, the string length for the C# expressed as a simple ratio would be 4/5 (.8), whereas in the 31-division system it is .7997, a difference that is vastly within the natural margin of error both acoustically and physically considering the minute imperfections of the string and fret material themselves. The next step in finding fret locations on the lute is to mark off distances from the nut for the desired pitches, but to do that, one must first decide which pitches to use. Just as one could theoretically have a plethora of split keys on the keyboard to account for all 31-divisions, one could place enough frets on the lute to account for every possible spelling of a note. There is no evidence that sixteenth-century players did this however. Rather, for practicality, they chose a particular range of the spiral of 1/4 comma fifths to situate their instrument in. This subset of pitches is sometimes called the “disposition” and can be defined by where the wolf 5th lies 19 . For practical reasons, the disposition of a lute is largely dependent on the tuning of its open strings. By this, I am not referring to any objective pitch reference but rather to what Wolfgang Freis calls “imagined tunings” 20 . This is a process familiar to any performer on a transposing instrument where fingerings are played as though in a particular key or mode. The functionalities of various pitches then are preserved, though the objective pitch standard is not. Considering this, lutes, can be “imagined” to have any pitch as their basis for the open strings, though, in practice, we find that sixteenth-century composers used only a limited subset of the possibilities. For simplicity’s sake, I have included charts for the pitches of the two most common “imagined tunings”, that of a lute in “A” and a lute in “G”. In practice, the “A” tuning would be appropriate for the vihuela, the baroque guitar (minus the 6th course), and the theorbo. The “G” tuning would be appropriate for the Renaissance lute, the archlute, and the so-called “English” theorbo. 19 Lindley, Mark. "Mean-tone." Grove Music Online. Oxford Music Online. 20 Freis, Wolfgang. “Perfecting the Perfect Instrument: Fray Juan Bermudo on the Tuning and Temperament of the ‘vihuela de mano’.” Early Music, Vol. 23, No. 3, Iberian Discoveries III (Aug., 1995). p. 425.
Figure 6. Fret positions for “A” tuning. In these figures one can see the functional difference between the “mi” frets and the “fa” frets. “Mi” frets function as leading tones and major thirds of triads in general and can be found two steps (minor semi-tone) in the 31-division scale above the natural pitch of the same same. “Fa” frets are one step of the 31-division above their “enharmonic equivalents” and can be found three steps (major semi-tone) above their lower diatonic neighbors. Figure 7. Fret positions for “G” tuning. The mode or key that a piece is written in will not only define what “imagined tuning” to use, but also what “disposition” to select. Clearly, figures 6 and 7 show that “mi” positions will favor dispositions toward the sharp end of the chain of fifths, whereas the fa positions will favor dispositions toward the flat end of the chain of fifths (see figure 1). The positions of frets II, III, IV, V, and VII are invariant since they create octaves and unisons with the open strings. Though Antonio Corona-Alcade has done an exhaustive analysis of the modes, tunings, and implied dispositions of Luis Milan’s El Maestro, choice of disposition for other sixteenth-century repertoire must currently by done on an ad hoc basis by the performer. Fortunately, a common practice for modal-polyphonic music on the lute and vihuela was to use a mode whose final is identical to one of the open strings. Thus for the “A” lute tuning one sees tonic triads of: A Major D Major G Major B Major E Major A minor D minor G minor B minor E Minor
Page 1 and 2: The Rule of Thirty-One: Realizing 1
Page 3 and 4: fifths which actually “overshoots
Page 5 and 6: fourths to avoid a discordantly wid
Page 7 and 8: implying, perhaps, that someone had
Page 9 and 10: and the Just Major 3rd, make this s
Page 11: Note Name Step Number (n) Frequency
Page 15 and 16: Notice how the marked “fa” and
Page 17 and 18: Figure 13. Template aligned to 6th
Page 19 and 20: Once the frets are placed, one can
Page 21: Freis, Wolfgang. “Perfecting the

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